FRG: Collaborative Research: Chern classes in Iwasawa Theory

目前已研究的岩泽理论的主要猜想是将岩泽模和Selmer复形的第一个陈类与p-进L级数联系起来。这个FRG项目的目标是将这一理论推广到更高的陈氏班级。这一概括的一个组成部分是关于如何以一种便于L系列研究的方式来定义更高的陈类。这将通过将Parshin和Beilinson的Adelic方法扩展到岩泽理论的背景来实现。推广的另一个组成部分是将更高的陈氏类不变量与L级数联系起来。要做到这一点，我们需要在算术问题中有足够的结构，以便使用L级数来了解其更高的余维特征。我们将考虑三种特殊情况：(I)全实域上的Greenberg猜想；(Ii)分裂素数上虚二次场的岩泽理论；(Iii)函数场的情形。关于(I)，Greenberg猜想自然岩泽模在余维一上有平凡支撑性；PI将利用L级数研究它们的余维二支撑性。关于(Ii)，Rubin、Kings和Johnson-Leung的工作建议人们应该通过与p-进L级数对相关的K_2群中的符号来研究第二类陈。关于(III)，PI将研究由Witte定义的类的Chern类映射下的映象，在函数域的情况下，在Iwa awa代数的高相对K-群内。这个项目的另一个组成部分是将证明第一类主要猜想所用的归约技巧推广到更高的类。这涉及到将群表示理论和研究傅里叶-Mukai函数中使用的倾斜复形和导出等价的理论推广到岩泽代数。这一建议涉及到关于代数方程的对称性群的基本问题。在20世纪50年代，S，岩泽开始了一种新的方法来研究这类方程，通过考虑它们在无穷族中的行为。岩泽证明了许多这样的族都有很好的渐近行为。这导致了关于由这些族产生的对称群的数字增长率的基本猜想。证明这类“主要猜想”是过去50年来抽象代数的中心目标之一。这一建议涉及对这些猜想的改进，这些猜想涉及更精确的增长率衡量标准。关于广泛的影响，关于这类代数问题的工作导致了对社会至关重要的技术的发展，例如改进了数据的压缩和安全传输。

The Main Conjectures of Iwasawa theory which have been studied up to now relate the first Chern classes of Iwasawa modules and Selmer complexes to p-adic L-series. The object of this FRG project is to generalize this theory to higher Chern classes. One component of this generalization concerns how to define higher Chern classes in a way that facilitates studying them by L-series. This will be done by extending to the context of Iwasawa theory the adelic methods of Parshin and Beilinson. Another component of the generalization has to do with connecting higher Chern class invariants to L-series. To do this, one needs enough structure in the arithmetic problem to see into its higher codimension features using L-series. Three particular cases will be considered are (i) Greenberg's conjecture over totally real fields, (ii) Iwasawa theory for imaginary quadratic fields at split primes, and (iii) the function field case. Concerning (i), Greenberg has conjectured that the natural Iwasawa modules have trivial support in codimension one; the PIs will study their codimension two support using L-series. Concerning (ii), work of Rubin, and of Kings and Johnson-Leung, suggests that one should study second Chern classes via symbols in K_2 groups associated to pairs of p-adic L-series. Concerning (iii), the PIs will study the images under Chern class maps of classes defined by Witte in the function field case inside the higher relative K-groups of Iwasawa algebras. One further component of this project has to do with generalizing to higher Chern classes the reduction techniques used in proving first Chern class Main Conjectures. This involves generalizing to Iwasawa algebras the theory of tilting complexes and derived equivalences which is used in group representation theory and in studying Fourier-Mukai functors.This proposal deals with fundamental questions about the groups of symmetries of algebraic equations. In the 1950's, Iwasawa began a new approach to the study of such equations by considering their behavior in infinite families. Iwasawa showed that many such families have well defined asymptotic behavior. This led to fundamental conjectures concerning the numerical growth rate of the symmetry groups arising from such families. The proof of such "Main Conjectures" has been one of the central goals of abstract algebra over the last 50 years. This proposal has to do with the refinements of these conjectures which deal with more precise measures of rates of growth. Concerning broad impacts, work on algebraic questions of this kind has led to the development of technology essential to society, such as the improved compression and secure transmission of data.